I need either a proof or a reference in modern (scheme-theoretic) language. According to Sansuc, this result can be gleaned from Borel's book on linear algebraic groups, but the old-style algebraic geometry language makes my head spin. Isn't there a modern proof of the above statement somewhere, for goodness sake?
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asked Jul 2, 2017 at 16:47
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1$\begingroup$ mathoverflow.net/questions/11322/… $\endgroup$– user19475Jul 2, 2017 at 17:04
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2$\begingroup$ Let $k$ be a field. Consider a split maximal $k$-torus $T$ in a connected reductive $k$-group $G$, the Borel $k$-subgroup $B\supset T$ such that the set of $T$-roots on ${\rm{Lie}}(B)$ is a chosen positive system of roots in $\Phi(G,T)$, and the Borel $k$-subgroup $B'\supset T$ "opposite" to $B$. If $U\subset B$ and $U'\subset B'$ are the $k$-split (!) $k$-unipotent radicals then multiplication $U'\times T\times U\to G$ is an open immersion with rational source. See Prop. 2.1.8(2),(3), 2.1.10, 2.2.9 in Pseudo-reductive Groups for a non-SGA3 proof based on scheme-theoretic dynamic methods. $\endgroup$– nfdc23Jul 2, 2017 at 17:18
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1$\begingroup$ An alternative unpublished reference for the same argument (punting some details to that book, but possibly easier to digest on a first pass) is given by Appendix A.2 and Theorem 6.1.1(2) in ams.org/open-math-notes/omn-view-listing?listingId=110663 $\endgroup$– nfdc23Jul 2, 2017 at 17:28
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3$\begingroup$ @TimoKeller: the reference you give addresses algebraically closed ground fields, but the OP seeks a reference applying over separably closed fields. For every imperfect field $k$ with characteristic $p>0$ (e.g., separably but not alg. closed fields) there exist unipotent smooth connected $k$-groups that are not $k$-rational (e.g., $y^p = x + a x^p$ for any $a \in k - k^p$, even if $p=2$), so the real miracle is that $\mathscr{R}_{u,k}(P)$ is always $k$-split (and descends $\mathscr{R}_u(P_{\overline{k}})$) for any parabolic $k$-subgroup $P$ in a connected reductive $k$-group $G$ for any $k$. $\endgroup$– nfdc23Jul 2, 2017 at 17:35
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$\begingroup$ Many thanks nfdc23. I think your comments answer my question (I have yet to check all the details, though, to convince myself that I understand you correctly) $\endgroup$– Cristian D. Gonzalez-AvilesJul 2, 2017 at 21:13
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2 Answers
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How's about these notes by Gille?
And these notes by Colliot-Thelene.
And these notes by Brian Conrad (Prop 7.2.3)
answered Jul 2, 2017 at 18:51
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2$\begingroup$ In section 11 on parabolic subgroups it is assumed that the ground field has char. 0. The "open cell" aspect is lurking inside the proof of 12.1(1) that rests as written on section 11, in particular 11.4 whose proof is referred to 4.10 in the Borel-Tits IHES paper on reductive groups that is presented in the non-scheme language (e.g., the part of interest in 4.10 of Borel-Tits rests on 4.7 there which only handles "intersection" in the sense of geometric points and "defined over $k$" aspects, so not set up as a scheme-theoretic intersection). So alas this doesn't seem to do the job. $\endgroup$– nfdc23Jul 2, 2017 at 19:11
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1$\begingroup$ @nfdc23 ok, look at the other set of notes. $\endgroup$ Jul 2, 2017 at 19:22
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2$\begingroup$ Alas, on page 9 of that alternative set of notes (with $G$ as in the statement of 4.12, so assumed reductive in positive characteristic) it is asserted in the lower part of that page that "we know $G$ becomes rational over a separable closure of $k$" and no reference is provided. Hence, it is assumed that the reader already knows the fact for which the OP is requested an alternative proof or reference for a proof. (I checked the entire document by searching for all occurrence of "rational" to confirm this is assumed knowledge of the reader, not proved therein.) Any other candidates? $\endgroup$– nfdc23Jul 2, 2017 at 19:58
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$\begingroup$ @nfdc23 You taunt me cruelly... I will look $\endgroup$ Jul 2, 2017 at 21:27
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1$\begingroup$ The 3rd link is the same as the set of notes which I linked (at the AMS website) in my 2nd comment to the OP. The Prop. 7.2.3 you point to is about unirationality, which is weaker than rationality (and holds over every field in the connected reductive case, whereas rationality does not -- though does over separably closed fields). In that comment of mine I pointed to another place in those same notes which does the job. $\endgroup$– nfdc23Jul 2, 2017 at 22:29
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This is 16.56 + 21.56 of J.S. Milne, Algebraic Groups: The theory of algebraic group schemes over a field, Cambridge U. P., 2017 (available September)
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1$\begingroup$ @CristianD.Gonzalez-Aviles: you are looking at a version on Milne's webpage, which is some older draft. The final version that is being published probably has a lot of revisions beyond that, including rearrangement of material and change of numerical labels; that would be why you don't find 16.56. It seems that "anon" has access to a version more recent than the one on Milne's website. $\endgroup$– nfdc23Jul 2, 2017 at 22:31
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$\begingroup$ Dear nfdc23, I think I'll include some version of your proof/suggested references as a remark in one of my future papers. That'll be good enough. Regarding your editing of my question, no problem. There's no God, anyway, so I'm cool with it. $\endgroup$ Jul 3, 2017 at 18:13